VI 3 Spectrum of compact operators Spectrum of

VI. 3 Spectrum of compact operators

Spectrum of T Let is called the resolvent set of T : spectrum of T

Eigenvalue and Eigenspace : eigenvalue of T the eigenspace associated If then

Remark 5 In general the inclusion is strict (except when there may exists ): such that and ( such belongs to an eigenvalue) but is not

Example Let then but

Proposition VI. 7 is compact and

Lemma 1. 2 Suppose that is a sequence consisting of totally different numbers such that then i. e. isolated elements. consists only

Theorem VI. 8 Let T is compact and Then (a) (b) (c) is finite or is a sequence tending 0.

Remark Given Then there is a compact operator T such that

VI. 4 Spectral decomposition of self-adjoint operators

Sesquilinear p. 1 Let X be a complex Hilbert space. is called sesquilinear if

Sesquilinear p. 2 B is called bounded if there is r>0 such that B is called positive definite if there is ρ>0 s. t.

Theorem 5. 1 The Lax-Milgram Theorem p. 1 Let X be a complex Hilbert space and B a a bounded, positive definite sesquilinear functional on X x X , then there is a unique bounded linear operator S: X →X such that and

Theorem 5. 1 The Lax-Milgram Theorem Furthermore exists and is bounded with p. 2

Self-Adjoint E=H is a Hilbert space Definition : if i. e. is called self-adjoint

Proposition VI. 9 T : self-adjoint, then

Remark of Proposition This Proposition is better than Thm VI. 7

Corollary VI. 10 Let then T=0 and

Propositions p. 1 Let be an orthogonal system in a Hilbert space X, and let U be the closed vector subspace generated by Let and be the orthogonal projection onto U where

Proposition (1)

Proposition (2)

Proposition (3) For each k and x, y in X

Proposition (4) For any x, y in X

Proposition (5) Bessel inequality

Proposition (6) ( Parseval relation) An orthonormal system is called complete and a Hilbert basis if U=X

Separable A Hilbert space is called separable if it contains a countable dense subset

Theorem VI. 11 H: a separable Hilbert space T: self-adjoint compact operator. Then it admits a Hilbert basis formed by eigenvectors of T.

VI. 1 Definition. Elementary Properties Adjoint

Lemma VI. 1 (Riesz-Lemma) Let For any fixed , apply Green’s second identity to u and then let in the domain we have